Pressure-driven fracture in elastic continuum materials

Experiments indicate that a monolayer of gas-liquid foam confined within a Hele-Shaw cell can exhibit brittle fracture when subject to an applied driving pressure. In this talk we characterise this brittle fracture mode by considering the propagation of an internally pressurised crack though a slab of elastic continuum material with low resistance to shear, extending the classical description of pressure-driven fracture in a linearly elastic material to a slab of finite-width. We employ a novel matched eigenfunction expansion approach to formulate the stress field, incorporating a global penalty term which we isolate by solving a Fredholm integral equation. We recover the well-known stress singularity in the neighbourhood of the crack tip, but demonstrate that the spatial extent of this stress field in the direction of the crack is set by the domain width irrespective of the shear modulus of the material. The versatility of this approach allows for considerable modifications in the physical properties of the fracturing material, including those characteristic of foams, where out-of-plane deflection of the structural elements and accompanying viscous resistance to motion over the plates of the Hele-Shaw cell are important. These modifications facilitate a solution of the continuum model in the limit of zero shear modulus, where the stress singularity is entirely absent and the lengthscale of the stress-field in the direction of the crack is instead set by the dissipation coefficients. We exploit this mis-match in lengthscales to construct an asymptotic description for a slender domain, analytically characterising the critical conditions for crack propagation as a function of the driving pressure and the domain width. Furthermore, we show that this outer asymptotic solution can be extended to describe materials with low but finite shear modulus, where the accompanying stress singularity around the crack tip is confined within a boundary layer adjacent to the crack surface.